sin^63x=(sin^23x)^3=(1-cos^23x)^3
d(cos3x)=(cos3x)`*dx
d(cos3x)=-3sin3xdx ⇒ sin3xdx=(-1/3)d(cos3x)
∫ sin^73xdx/cos^63x=(-1/3) ∫ (1-cos^23x)^3d(cos3x)/(cos^23x)^3 =
=(-1/3) ∫ (1/cos^23x)-1)^3d(cos3x)=
=(-1/3) ∫ ((1/cos^63x)-(3/cos^43x)+(3/cos^23x)-1)d(cos3x)
=(-1/3)*(cos^(-5)3x)/(-5) +(cos^(-3)3x)/(-3) -(cos^(-1)3x)/(-1)+(1/3)cos3x + C=
=(1/15)*(1/cos^53x)-(1/3)*(1/cos^33x)+(1/cos3x)+(1/3)cos3x+C
О т в е т.
(1/15)*(1/cos^53x)-(1/3)*(1/cos^33x)+(1/cos3x)+(1/3)cos3x+C